Yes, ESP8266's are a great solution for Arduino web connectivity. Lots of tutorials exist for them. It's a little annoying how they communicate over serial, when the '328p only has one serial port, which is also used for programming it. So you'll have to use software serial, or disconnect the wifi while uploading the sketch to the Arduino. I'd have the arduino running a simple server that can return status, and take commands.
For phones, you could also do bluetooth - but those modules also use serial, which starts to get awkward if you also want WiFi (if you're using '328p based board - if you're using one based on something with more than one uart, this of course doesn't apply). I'd probably just use the wifi/http server method though.
You should also figure out what you want your device to do - presumably you want it to interact with the coffee maker somehow. For example, you might want temperature sensors, to see if the coffee is still warm (these are also an "easy" way to find out when the machine is running without having to touch line voltage). Or light sensors (maybe you could rely on the fact that coffee is dark to detect coffee level?). Or motion sensors, to alert you to someone taking the last cup? And you'll probably want an LED to display status.... Don't neglect these things in focusing on connectivity; you've still got to wait out a shipping delay.